Difference between revisions of "2014 USAJMO Problems"
(→Problem 1) |
(→Problem 3) |
||
| Line 8: | Line 8: | ||
[[2014 USAJMO Problems/Problem 2|Solution]] | [[2014 USAJMO Problems/Problem 2|Solution]] | ||
===Problem 3=== | ===Problem 3=== | ||
| + | Let <math>\mathbb{Z}</math> be the set of integers. Find all functions <math>f : \mathbb{Z} \rightarrow \mathbb{Z}</math> such that <cmath>xf(2f(y)-x)+y^2f(2x-f(y))=\frac{f(x)^2}{x}+f(yf(y))</cmath> for all <math>x, y \in \mathbb{Z}</math> with <math>x \neq 0</math>. | ||
[[2014 USAJMO Problems/Problem 3|Solution]] | [[2014 USAJMO Problems/Problem 3|Solution]] | ||
, 
, 
be real numbers greater than or equal to 
. Prove that ![\[\min{\left (\frac{10a^2-5a+1}{b^2-5b+1},\frac{10b^2-5b+1}{c^2-5c+10},\frac{10c^2-5c+1}{a^2-5a+10}\right )}\leq abc\]](http://latex.artofproblemsolving.com/3/a/d/3ad6f9ab816469889e5b8452f203c0e9b01647f4.png)
be the set of integers. Find all functions 
such that ![\[xf(2f(y)-x)+y^2f(2x-f(y))=\frac{f(x)^2}{x}+f(yf(y))\]](http://latex.artofproblemsolving.com/9/f/d/9fdb11dbf102dfe0783b1d31466e276420e3913d.png)
for all 
with 
.
